These are good questions. I'll get to the core of them after a one-paragraph rant about the aggregator linked at the top, and three paragraphs about how this bit of "science communication" (s̶c̶a̶r̶e̶ sarcasm quotes) means no answers for you about this particular visualization. I did end up peppering in a couple relevant bits of information in them though. In case it's not clear, the ordering of paragraphs below is not the order in which they were written.

Preliminarily: phys.org is hot garbage. It mostly reproduces institutions' (universities, NASA) press releases with its own ads. Rarely there is original content of dubious quality. This is not the latter, it's just a NASA-written blurb. Fortunately the same content is at the original source https://science.nasa.gov/supermassive-black-holes/new-nasa-b... -- absent that I probably wouldn't have started an answer.

Unfortunately the NASA link and its link to youtube give too little information to say anything reliable about the numerical relativity (NR) "codes" (s̶c̶a̶r̶e̶ jargon quotes) for this particular visualization. Yes, it's pretty and cool; yes it will excite relativists as well as laypersons; however, how about tossing the former a little drop of technical information? The supercomputer used and how much time was spent on it is not really useful to know.

I can only guess that the visualizer, Jeremy Schnittman, would want to generate data using the tools with which he's most familiar. Digging around in his publication history <https://scholar.google.com/citations?user=MiUTIQwAAAAJ> I see that he uses lots of Monte Carlo methods, often averaging over many very slightly different simulations (which might individually be wrong). I can see that for his black hole related work (mostly studying X-rays flying about just outside black holes, but also other phenomena close to but outside the horizon, or a little before the most strongly relativistic parts of black hole mergers) he prefers his own (Monte Carlo) tool Pandurata, which is described in Schnittman & Krolik 2013 <https://iopscience.iop.org/article/10.1088/0004-637X/777/1/1...>.

It is not at all obvious to me, especially given the scant information provided by NASA publicity, how he would safely apply these sorts of toolsets to the interior black hole metric. We aren't even told what the metric is, really; I assume Kerr with modest angular momentum, given his previous publications focus on astrophysically-reasonable Kerr black holes. There's another hint in what in the video looks like dimming at the receding limb (on the right) of the accretion disc. But this visualization could also be just Schwarzschild. Who knows? I look forward to his own professional writeup!

[Other NR tools are available, e.g. <https://nrpyplus.net/> and <https://grchrombo.org/movies/>]

Consequently I'll focus in on this part of your comment:

> these kinds of scales [or] galaxy simulations [...]

This is in the realm of numerical relativity and computational astrophysics respectively. There is an overlap.

Although I had in mind a couple resources about your questions, they're mostly textbooks which aren't freely available. So I first visited Sebastiano Bernuzzi's always useful syllabus http://sbernuzzi.gitpages.tpi.uni-jena.de/nr/ (nr is for Numerical Relativity, solving Einstein's equations with computers) and picked out two useful freely-available resources to start with. They are both called "lecture notes" but are really mini textbooks. They both have excellent bibliographies.

Choptuik's 2006 "Numerical Analysis for Numerical Relativists" (PDF) http://laplace.physics.ubc.ca/People/matt/Teaching/06Mexico/... is awesome. It focuses on finite difference techniques, which dominate in numerical relativity, particularly where black holes are concerned.

Like many others, Bernuzzi's 2021 3+1 Numerical Relativity <http://sbernuzzi.gitpages.tpi.uni-jena.de/nr/notes/2021/main...> points to it in section 2.4, where you will find references to other and newer treatments of various numerical relativity methods. Other techniques get used too, finite-element, for example. For galaxy stuff, you would want a resource on e.g. smoothed particle hydrodynamics or particle-particle/particle-mesh-Ewald.

ETA^2: wow, an actually useful physics SE q&a on that last bit (contrasting finite difference & finite element methods for black holes) from a little less than two years ago (direct link to imho good answer): <https://physics.stackexchange.com/a/725998>

ETA: this is more just a bookmark for me. Even though, like Chopotuik above, it's 17 years old, <https://www.cita.utoronto.ca/~pfeiffer/talks/07Apr_Jacksonvi...> is a really great slide deck.

ETA^3: Bernuzzi "Introduction to Numerical Relativity" @ IHÉS winter 2024 https://www.youtube.com/watch?v=RcdntEBrcuM is probably pretty good.